introduction to coding dsge models with dynare

Dynare

Introduction to Coding DSGE Models withDynare

Macroeconomic Theory (MA1314)

Juha Kilponen

Bank of Finland

Additional Material for Lecture 3

Dynare

DynareI Dynare is preprocessor and a collection of Matlab (and GNUOctave) routines, which solves DSGE models

I Dynare can also be used to estimate DSGE Models and evenMarkov Switching Bayesian VAR models

I The Dynare home page is http://www.dynare.org. You candownload dynare from here. Latest version is 4.4.1.

I The Dynare manual is available athttp://www.dynare.org/documentation-and-support/manual

I Instructions how to install Dynare can be found in the manual(Chapter 2, Installation and configuration)

I See also the Dynare Tutorialhttp://www.dynare.org/documentation-and-support/manual

I WIKI inhttp://www.dynare.org/documentation-and-support/wiki is auseful user forum

Dynare

The structure of a typical Dynare file

I You tell Dynare what the variables, shocks and parameters(+parameter values) of your model are

I You write down the model equationsI You tell Dynare to find the steady state and check stabilityconditions

I Dynare linearizes the model around the steady stateI ... and solves the recursive equilibrium laws of motion of thelinearized model

I The model is analyzed via impulse responses, stochasticsimulations and moments

Dynare

The structure of a typical Dynare fileI var;

I Define the ’endogenous’variables of your model

I varexo;I Define the list of shocks in your model

I parameters;I Define the list of parameters and then assign the parametervalues

I you may also find it useful to define some parametertransformations

I e.g. beta is a parameter, and you define steady state realinterest rate rr = 1/beta −1

I more generally, you may give the steady state values ofendogenous variables in terms of parameters

I "var" and "parameters" - blocks end with command "end";

Dynare

The structure of a typical Dynare file: Writing down themodel

I The model block starts with

model;

I ... and ends withend;

I Between ’model;’and ’end ;’you write down the necessaryequations defining the dynamic equilibrium of the model

I first-order conditions, constraints and any additionaltransformations of variables

I Remember: you need as many equations as there areendogenous variables

I Certain functions are allowed with model command, seeManual for details

Dynare

I Time indices are given in parenthesisI Xt+1 is written X (+1), (These are forward looking variables)I Xt−1 is written X (−1)I Xt is written X (no time index needed for the current period)

Dynare

Timing Conventions Cnt’dI In Dynare, the time index refers to the period when the valueof the variable is determined

I The value of the capital stock, which is used in production inperiod t, is predetermined in period t !

I In period t − 1, the agents decide how much to consume andinvest, and they simultaneously determine the size of thecapital stock that will be available in period t

I In a theory model we write the period t production function

Yt = ZtK αt L1−αt

... but in Dynare we would write

Y = Z ∗K(−1)alpha ∗ L(1−alpha)

I Alternatively, you can declare a particular variable aspredetermined using command "predetermined_variables".

Dynare

Log Linearisation vrs. Linearisation

I Dynare linearizes the model around the steady stateI Remember the advantages of log-linearizations: We aredealing with percentage deviations from the steady state (orthe balanced growth path)

I Log-deviations (or percentage deviations) are easy to interpret(is a deviation ’small’or ’large’?)

I Similar measures are used in empirical examination of the data(e.g. applying HP filter to log(GDP))

I To obtain log linear solution, you can write down the model interms of logarithmic transformations of the original variables

I When Dynare linearizes the model in terms of the logarithmictransformations, it log-linearizes the model in terms of theoriginal variables

Dynare

ExampleI Adopt the notation

ly = log(Y ), lc = log(C ), lk = log(K ) etc.

I Note: For employment we use the notation

lh = log(L)

where h refers to hours worked (the alternative ll just doesnot look vey nice)

I Also remember the Dynare conventions pertaining to timeindexation

I Then the resource constraint

Yt = Ct +Kt+1 − (1− δ)Kt

can be written as

exp(ly) = exp(lc) + exp(lk)− (1− delta) ∗ exp(lk(−1))

Dynare

Expectations operatorI The expectation operator Et is not used in Dynare code.

I Dynare ’knows’when one has to take expectations, we do nothave to tell this explicitly.

I Thus the consumption Euler equation

Et

[Ct+1βCt

]= Et

[αZt+1

(Kt+1Lt+1

)α−1+ 1− δ

]or

Et

[Ct+1βCt

]= Et

[αYt+1Kt+1

+ 1− δ

]is written as1beta

∗ exp(lc(+1)− lc) = alfa ∗ (exp(y(+1)− lk) + 1− δ

I Notice: lh = log(L) (h refers to hours worked)I Also notice: In Dynare code, you need the multiplication sign ∗

I Extension: Dynare allows you to change the information setavailable for agents. For instance, writingEXPECTATION(-1)(C(+1)) mean agents use time t-1information to take expectation of variable C for time t+1

Dynare

Finding the steady stateI It is often necessary to give Dynare an initial guess of thesteady state values, after the command

initval;

I Can be combined with command "endval;" and simulate themodel’s adjustment between initial and endvalues. Typicallyused in deterministic simulations (command: simul;).You tell Dynare to find the steady state with the command

steady;

I Different algorithms to solve the steady state exist, for detailssee the Manua.

I To check the system is stable, you can use command

check;

I Checks the Blanchard Kahn conditions.

Dynare

Characterizing the shock structure of the model

I shocks; Shock characterization block startsI var e;

I Here we want to shock only one ’variable’(the TFP shock e)

I stderr 0.007;I The standard error of the shock is 0.007I If there are several shocks, say e_z and e_c, you have to bemore careful

I stderr e_z 0.007; stderr e_c 0.002;

I More generally, if there are several shocks, you may want togive the variance-covariance matrix of the shocks; see DynareManual

I end; Shocks charaterisation ends

Dynare

Analyzing the model via impulse responses, stochasticsimulations and moments

I stoch_simul(order=1,irf=20) ly lh lc li lk z;I order=1 means that Dynare takes a first order Taylorapproximation around the steady state (order=2 => secondorder approximation, order=3 => third order approximation

I irf=20 requests impulse responses for 20 periods. Default is 40I Dynare also computes the moments of the endogenousvariables

I mean, standard deviation, variance, skewness, kurtosisI (contemporaneous) covariance matrixI autocorrelations (default: up to the 5th order)

I The list of variables, we want to Dynare to analyze is ly lh lc lilk z

I Notice that z ≡ log(Zt )− log(Z)= log (Zt ) , since Z = 1

and log(Z)= 0 (a useful normalization).

Dynare

Analyzing the model via impulse responses, stochasticsimulations and moments

Alternatively we can write:

I stoch_simul(order=1,periods=1000,irf=20) ly lh lk lc lw z;I Now Dynare simulates the model for 1000 periods, andcomputes the moments based on the simulated data

I Or, stoch_simul(order=1,hp_filter=1600, irf=20) ly lh lk lclw z; Now we HP filter the variables before computing themoments

I For additional options in stoch_simul - command, see DynareManual

Dynare

The structure of the code in a nutshellvar (list of endogenous variables );varexo (list of shocks) ;parameters (parameters + parameter values + transformations);

model;model equations;end;

initval; (initial guesses for computing the steady state)steady; (compute the steady state)

shocks; (the shock structure of the model )var (Variables to be shocked);stderr (the standard error of the shocks);end;stoch_simul(order=1,irf=100) ly lh lc li lk z; (analyzing themodel)

Dynare

Hansen’s Indivisible Labour RBC model: Equilibriumequations

1 = βEt

[(CtCt+1

)(αZt

(Kt+1Lt+1

)α−1+ 1− δ

)]

ψCt = (1− α)Zt

(KtLt

)α

Yt = Ct +Kt+1 − (1− δ)KtYt = ZtK α

t L1−αt

I = Yt − CtWt = (1− α)

YtLt

logZt = (1− φ) logZ + φ logZt−1 + εt , εt is iid

Dynare

Dynare file: Hansen’s RBC model

// following are the ’endogenous’variables:var ly, lh, lc, li, lw, lk, z ;

// following are the shocksvarexo e;

parameters beta, fi, alpha, delta, psi;

alpha = 0.33;fi = 0.74;beta = 0.99;delta = 0.024;psi =0.262;

Dynare

Dynare file: Hansen’s RBC model

// Steady state expressed in terms of parametersrr=1/beta-1;zss=1;css=(1-alpha)*(zss/psi)*(alpha*zss/(rr+delta))^(alpha/(1-alpha));kss=css/((rr+delta)/alpha-delta);hss=(1/psi)*(1-alpha*rr/(rr+(1-alpha)*delta));yss= zss*kss^(alpha)*hss^(1-alpha);iss=delta*kss;wss=(1-alpha)*yss/hss;

Dynare

Dynare file: Hansen’s RBC modelmodel;

(1/beta)*exp(lc(+1)-lc)= alpha*exp(ly(+1)-lk) + 1 - delta;//Euler equation

psi*exp(lc)= (1-alpha)*exp(ly-lh); //Labour market equilibrium

exp(ly) = exp(lc) + exp(lk) - (1-delta)*exp(lk(-1)); //Resourceconstraint

exp(ly) = exp(z+alpha*lk(-1)+(1-alpha)*lh); //Productionfunction

exp(li)=exp(ly)-exp(lc); // Investment

exp(lw)=(1-alpha)*exp(ly-lh); //Real wage

z = fi*z(-1) + e; //Law of motion of TFP

end;

Dynare

Dynare file: Hansen’s RBC model

//Initial guesses for the computation of steady state

initval;lc=log(css);lh=log(hss);lk=log(kss);ly=log(yss);li=log(iss);lw=log(wss);z=zss;end;

steady;

Dynare

Dynare file: Hansen’s RBC model

shocks;var e;stderr 0.007.;end;

stoch_simul(order=1,irf=100) ly lh lc li lk z;

Dynare

Saving and running the Dynare file

I Save the file as [Name of the file].modI For example rbc.mod

I Run the model by writing in the Matlab command window

dynare rbc.mod

I You may also run the Dynare file from a separate m.-file (thisis practical if you, say, want to solve the model many times,with different parameter values)

Dynare

I An alternative way to write down the model.I You write down the model in terms of the original variables(not log-transformations)

I Then you add the log-transformations as extra variables

Dynare

Dynare file: Hansen’s RBC model (alternative form)

// following are the ’endogenous’variables:var Y, L, C, I, W, K, ly, lh, lc, li, lw, lk, z ;

// following are the shocksvarexo e;

parameters beta, fi, alpha, delta, psi;

alpha = 0.33;fi = 0.74;beta = 0.99;delta = 0.024;psi =0.262;

Dynare

Dynare file: Hansen’s RBC model (alternative form)

I Steady state expressed in terms of parametersI Exactly the same as above, in the main version

Dynare

Dynare file: Hansen’s RBC model (alternative form)model;

(1/beta)*C(+1)/C= alpha*Y(+1)/K+ 1 - delta;//Euler equation

psi*C= (1-alpha)*Y/H; //Labour market equilibrium

Y = C + K - (1-delta)*K(-1); //Resource constraint

Y = exp(z)*K(-1)^alpha*L^(1-alpha); //Production function

I=Y-C; // Investment

W=(1-alpha)*Y/L; //Real wage

z = fi*z(-1) + e; //Law of motion of TFP

Dynare

Dynare file: Hansen’s RBC model (alternative form)// model code continues

// define log transformations of the endogenous varialbes

ly=log(Y);

lh=log(L);

lc=log(C);

li=log(I);

lw=log(W);

lk=log(K);

end;

Dynare

Dynare file: Hansen’s RBC model (alternative form)//Initial guesses for the computation of steady state

initval;C=css;L=hss;K=kss;Y=yss;I=iss;W=wss;lc=log(css);lh=log(hss);lk=log(kss);ly=log(yss);li=log(iss);lw=log(wss);z=zss;end;

steady;

Dynare

Dynare file: Hansen’s RBC model (alternative form)

shocks;var e;stderr 0.007.;end;

stoch_simul(order=1,irf=100) ly lh lc li lk z;//Notice: Here you ask Dynare to analyze the impulse responses//and the moments of the log-transformations.

Dynare

Output

Output includes

I Policy and transition functions of the (log)linearized modelI Moments of the endogenous variables

I mean, variance, standard deviation, skewness, kurtosisI matrix of contermporaneous correlationsI coeffi cients of autocorrelation

I Impulse responsesI If if you have included the periods option in

stoch_simul(order=1,periods=1000,irf=100) ly lh lc li lk z;

the output also includes the results (time paths of endogenousvariables) from the stochastic simulation.

Dynare

Output

I The output is shown on the screen, and in separate figures(impulse responses).

I Output is also stored in a separate structure, called oo_.Other information of the model is saved in structure M_

I The structure oo_ contains (for example)I The steady state (oo_.steady_state )I The variance-covariance matrix (oo_.var)I The autocorrelations (oo_.autocorr)I The impulse responses (oo_.irfs)I The coeffi cients of the policy and transition functions (oo_.dr)I Results (time paths of endogenous variables) from stochasticsimulations (oo_.endo_simul)

I Structure M_ contains add information of the model object.

Dynare

Output: Policy and transition functions

ly lh lc li lk zConstant 0.019 −1.1 −0.24 −1.44 2.29 0lk(−1) −0.0075 −0.50 0.50 −1.67 0.94 0z(−1) 1.87 1.68 0.18 7.4 0.18 0.74e 2.52 2.28 0.25 10.0 0.24 1

I Transition functions: how the period t values of the statevariables (lk and z) depend on the period t − 1 values of thestate variables, and the shock

I Policy functions: how the period t values of the othervariables depend on the period t − 1 values of the statevariables, and the shock

Dynare

Output: Policy and transition functions

ly lh lc li lk zConstant 0.019 −1.1 −0.24 −1.44 2.29 0lk(−1) −0.0075 −0.50 0.50 −1.67 0.94 0z(−1) 1.87 1.68 0.18 7.4 0.18 0.74e 2.52 2.28 0.25 10.0 0.24 1

I Rows: period t values of the variablesI Columns: period t − 1 values of the state variables + theshock (e)

I The term ’constant’is the steady state (of log transformation)I E.g. lh = −1.1 and thus L = exp(−1.1) = 0.33

Dynare

Output: Policy and transition functions

ly lh lc li lk zConstant 0.019 −1.1 −0.24 −1.44 2.29 0lk(−1) −0.0075 −0.50 0.50 −1.67 0.94 0z(−1) 1.87 1.68 0.18 7.4 0.18 0.74e 2.52 2.28 0.25 10.0 0.24 1

I The entries of the table give the coeffi cients of the(log)linearized policy and transition functions. For example(the first column)

ly = 0.019︸ ︷︷ ︸ly

− 0.0075∗(lk (−1)− lk

)+ 1.87∗ z(−1)+ 2.52∗ e

Dynare

Output: Connection to our earlier results (Lecture 2)

ly = 0.019︸ ︷︷ ︸ly

− 0.0075 ∗(lk (−1)− lk

)+ 1.87 ∗ z(−1) + 2.52 ∗ e

(1)I How to write (1) in terms of log deviations yt , kt and zt?I Remember that

ly − ly = log(Yt )− log(Y)= yt ;

lk (−1)− lk = log(Kt )− log(K ) = ktI Also notice that

2.52 ∗ z = 2.52 ∗φ︷︸︸︷0.74︸ ︷︷ ︸

1.87

∗ z(−1) + 2.52 ∗ e

I Thus (1) can be rewritten as

yt = −0.0075 ∗ kt + 2.52 ∗ zt

Dynare

Output: Connection to our earlier results (Lecture 2)

ly lh lc li lk zConstant 0.019 −1.1 −0.24 −1.44 2.29 0lk(−1) −0.0075 −0.50 0.50 −1.67 0.94 0z(−1) 1.87 1.68 0.18 7.4 0.18 0.74e 2.52 2.28 0.25 10.0 0.24 1

I More generally, in terms of the notation presented in Lecture2, the transition and policy functions can be re-expressed asfollows

kt+1 = 0.94 ∗ kt + 0.24 ∗ zt ; ct = 0.50 ∗ kt + 0.25 ∗ ztht = −0.50 ∗ kt + 2.28 ∗ zt ; ιt = −1.67 ∗ kt + 10.0 ∗ zt

Dynare

Solving recursive stochastic linear systems - general caseI In general, analysis of DSGE models involves a solution to thefollowing system

Et (f (yt+1, yt , yt−1, zt ) = 0 (2)

I yt is a vector of endogenous state variables, zt is a vector ofexogenous stochastic shocks. It is often useful to writezt = σεt , E (εt ) = 0, E (εtε′t ) = Σε. σ is then scale variableand you can assume that auxiliary shocks εt has unit variance.

I Notice that f is known.I Policy (solution) function is

yt = g(yt−1, zt , σ)

Thus,

yt+1 = g(yt , zt+1, σ) = g(yt , zt+1, σ) = g(g(yt−1, zt , σ), zt+1, σ)(3)

Dynare

Solving recursive stochastic linear systems cnt’dI Using (3) in (2), we find that

Et{F (yt−1, zt , zt+1, σ)} = 0 (4)

I A deterministic steady state y satisfies f (y , y , y , 0) = 0 andy = g(y , 0, 0). This is nothing but a non-linear system whichcan easily be solved. (e.g. Matlab, fsolve function). The onlyproblem is that there can be several solutions. We need tochoose only one for approximation.

I There are different methods to deal with (4). For instance,Dynare (1st order approx) computes approximated decisionrules of the form

yt = y + gy yt−1 + gzzt , with y = yt − yI It all eventually collapses into a problem of finding y , gy andgz , such that the system has a saddle path property.

Dynare

Dynare - first order approximation around y:

Et{F (yt−1, zt , z′, σ)}= (fy ′gygy + fy0gy + fy_)y

+(fy ′gygz + fy0gz + fz )z (5)

+(fy ′gygσ + fy0gσ)σ

= 0

where

y = yt−1 − y , z = zt , z ′ = zt+1fy ′ =

∂f∂y ′, fy0 =

∂f∂y0

, fy_ =∂f

∂yt−1,

fz =∂f∂z, gz =

∂g∂z, gσ =

∂g∂σ, gy =

∂g∂yt−1

I Basic idea: Equation (5) needs to hold for any arbitrary y , zand σ.

Dynare

Dynare - first order approximation cnt’d

I We have the following 3 matrix equations

(fy ′gygy + fy0gy + fy_) = 0

(fy ′gygz + fy0gz + fz ) = 0

(fy ′gygσ + fy0gσ) = 0

I From the last one, we can already deduce that gσ = 0. This isa certainty equivalence property.

I But we have still 2 unknowns gy , gz . We recover gz from thefirst equation and gz from the second.

I The hardest one to solve is gy .

Dynare

Solving

gz

(fy ′gygz + fy0gz + fz ) = 0

gz = −(fy ′gy + fy0)−1fz

Dynare

Solvinggy

I Consider following state space representation[0 fy ′I 0

] [Igy

]gy y =

[−fy_ −fy00 I

] [Igy

]y (6)

or [0 fy ′I 0

]︸ ︷︷ ︸

D

[yt − yyt+1 − y

]︸ ︷︷ ︸

xt+1

=

[−fy_ −fy00 I

]︸ ︷︷ ︸

E

[yt−1 − yyt − y

]︸ ︷︷ ︸

xt

(7)

I So, above can be expressed as

Dxt+1 = Ext (8)

Dynare

Generalized Schur decomposition

I We want stable solution to (8).I Problem of finding a solution to (8) is that D might not beinvertible. But to solve the system we still need to find itseigenvalues and eigenvectors. The generalized Schurdecomposition allows to solve generalised eigenvalues.

I There exist matrices Q,T,Z,S such that

D = QTZ

E = QSZ

I where T and S are upper triangular, Q ′Q = I , Z ′Z = I .

Dynare

Applying the decompositionD[Igy

]gy y = E

[Igy

]y[

T11 T120 T22

] [Z11 Z12Z21 Z22

] [Igy

]gy y

=

[S11 S120 S22

] [Z11 Z12Z21 Z22

] [Igy

]y

I To exclude explosive trajectories we impose

Z21 + Z22gy = 0

gy = −Z−122 Z21

I A unique stable trajectory exists if Z22 is non-singular: thereare as many roots larger than one in modulus as there areforward looking variables in the model (Blanchard and Kahn(1980) condition) and the rank condition is satisfied.

Dynare

First order approximated decision function and variance

yt = y + gy yt−1 + gzztE (yt ) = y

Σy = gyΣyg′y + σ2guΣεg ′u

I Variance Σy is solved using an algorithm for Lyapunovequations. Second moments can therefore be computeddirectly, without a need for simulation. The same applies toAR coeffi cients.